ITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR
|
|
- Charlene Simpson
- 5 years ago
- Views:
Transcription
1 Fixed Poit Theory, 14(2013), No. 1, odeacj/sfptcj.htl ITERATIONS AND FIXED POINTS FOR THE BERNSTEIN MAX-PRODUCT OPERATOR MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN,,, Departet of Matheatics ad Coputer Sciece Uiversity of Oradea, Uiversitatii 1, , Oradea, Roaia E-ail: E-ail: E-ail: E-ail: Abstract. I this paper we study the sequece of successive approxiatios, the fixed poits ad the Ishikawa iterates for the Berstei ax-product operator. Key Words ad Phrases: Berstei ax-product operator, oexpasive operator, sequece of successive approxiatios, fixed poits, Ishikawa iteratios Matheatics Subject Classificatio: 41A36, 47H09, 47H10, 47H Itroductio For a fuctio f : 0, 1] R + (here x R + eas x 0), the Berstei axproduct approxiatio operator was for the first tie defied (ad forally studied) i 7], pp , by the forula (f)(x) = p,k (x)f ( ) k, p,k (x) where p,k (x) = ( ) k x k (1 x) k ad p,k(x) = ax k={0,...,} {p,k (x)}. Notice that the Berstei ax-product operator is obtaied fro the liear Berstei polyoial writte i the for B (f)(x) = p,k(x)f(k/) p,k(x) ad replacig here the su operator by the axiu operator. Surprisigly, with respect to the classical Berstei polyoials, i the whole class of cotiuous fuctios o 0, 1], the ax-product Berstei operators do ot loose the approxiatio properties. Moreover, they preset the advatage that for large classes of fuctios iprove the order of approxiatio to the Jackso-type order. 39
2 40 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I ore details, it was proved i 3], 4] that B is a oliear (ore exactly subliear o the space of positive fuctios) operator, well-defied for all x R, ad a piecewise ratioal fuctio o R. Also, i 3] it was proved that B possesses soe iterestig direct approxiatio results ad shape preservig properties. For exaple, while i geeral the order of uifor approxiatio was foud to be ω 1 (f; 1/ ) 0,1], however, for soe subclasses of fuctios icludig for exaple the class of cocave fuctios ad also a subclass of the covex fuctios, the order of approxiatio is essetially better, aely is ω 1 (f; 1/) 0,1]. I additio, i 3] it was proved that B (f) is cotiuous for ay positive fuctio f, preserves the ootoicity ad the quasicovexity of f. For strictly positive fuctios, iproved direct approxiatio results by the Berstei ax-product operator we have obtaied i 5]. For the classical Berstei polyoials B (f)(x), i the paper of Rus 11] the wellkow Kelisky Rivli s result i 10] statig that for all f C0, 1], x 0, 1] ad N it holds li B (f)(x) = f(0) + f(1) f(0)]x = B 1 (f)(x) (here B (f) deotes the th iterate of the sequece of successive approxiatios), is proved i a very siple ad elegat aer, by usig the Baach fixed poit theore. Note here that B 1 (f)(x) = f(0) + f(1) f(0)]x is a fixed poit for the operator B. Also, if = depeds o ad if li = 0, the it is kow that (see e.g. 10]) li B (f)(x) = f(x) uiforly i 0, 1]. Siilar studies for the iterates of other kids of Berstei-type operators were obtaied via fixed poit theory i e.g. Agratii 1], Rus 12] ad Agratii-Rus 2]. The ai ai of the preset paper is to ake a siilar study for the iterates of. It is worth otig that due to the fact that B is ot a cotractio (is oly a o-expasive operator), the ethods used i the case of Berstei polyoials caot be used for the Berstei ax-product operators, so that ew ethods are required. The pla of the paper goes as follows. Although the Berstei ax-product operator is ot a cotractio, as a aalogue of the above etioed Kelisky Rivli s results for the Berstei polyoial, i Sectio 2 of the preset paper firstly we prove by a direct ethod that for ay fixed N ad f : 0, 1] 0, + ), the sequece of successive approxiatios of the the Berstei ax-product operator ] (f)(x), still uiforly coverges for to a fixed poit of B. Also, the liits of the double sequece (a (f)), N for other iterdepedeces betwee ad are calculated. I the sae sectio, iportat subsets of the set of fixed poits of the operator B are cocretely deteried. Fially, i Sectio 3 we study the covergece of so-called Ishikawa iterates for the operator B. oliear operator, deoted by a (f)(x) = 2. The sequece of successive approxiatios ad fixed poits for For the proof of the covergece of the sequece of successive approxiatios of, we eed the followig three auxiliary results.
3 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 41 The first result obtaied oe refers to the fact that ulike the classical Berstei (liear) operator B (f) which is a cotractio, the ax-product Berstei (oliear) operator B (f) is oly a oexpasive operator. This eas that the Baach fixed poit theore caot be applied i this case. Theore 2.1. For ay N, the ax-product Berstei operator B : C + 0, 1] C + 0, 1] is oexpasive, that is B (f) B (g) f g, for all f, g C + 0, 1], where C + 0, 1] = {f : 0, 1] R + ; f is cotiuous o 0, 1]}, R + = {x R; x 0} ad deote the uifor or i C + 0, 1]. Proof. We easily get (f)(x) B (g)(x) p,k(x)f(k/) p,k (x)g(k/) p,k(x) f g, which proves the theore. Rearks. 1) I geeral, the iequality i Theore 2.1 is ot strict, that is there exists f, g C + 0, 1], such that B (f) B (g) = f g. Ideed, let us choose, for exaple, f oicreasig o 0, 1] ad g = 0 o 0, 1]. By Corollary 5.6 i 3], it follows that B (f) is also oicreasig o 0, 1], which iplies that f = f(0), B (f) = B (f)(0) ad by the obvious relatioship B (f)(0) = f(0), it iplies B (f) B (g) = B (f) = f(0) = f = f g. 2) Note that Lea 2.5 i 6] shows that for ay bouded f : 0, 1] R + ad N, B (f) Lip L 1, with L = C 2 f, C > 0 beig a costat idepedet of f ad, where Lip L 1 = {f : 0, 1] R; f(x) f(y) L x y, for all x, y 0, 1]}. I the ext result we obtai a explicit value for C i the above Reark 2. Theore 2.2. For all f C + 0, 1] ad h 0 we have ω 1 (B (f); h) 6πe 2 2 f h, where ω 1 (f; h) = sup{ f(x) f(y) ; x, y 0, 1], x y h} deotes the odulus of cotiuity. Proof. Aalysig the proof of Lea 2.5 i 6], we get ω 1 (B (f); h) 1 2 f h, c 2 1 where it is easy to observe that the costat c 1 > 0 (idepedet of x ad ) coes fro Lea 2.4 i 6] as satisfyig the iequality p,k(x) c1, for all x 0, 1] ad N. Aalysig ow the proof of Lea 2.4 i 6], it easily follows that c 1 = c 2 1 e, where c 2 > 0 is ow the costat that appear i the stateet of Lea 2.3 i 6] as satisfyig { j i p,j ( + 1 ), p,j( j + 1 } + 1 ) c 2, for all N, ad j {0, 1,..., }, where c 2 > 0 is a absolute costat idepedet of ad j.
4 42 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I cotiuatio, aalysig the proof of Lea 2.3 i 6] ad deotig A = (2!) 2 1 (2)! 2+1, sice li A = π 2 ad because it is easy to prove that (A ) is icreasig, we get 2 π < A <, for all N. 3 2 This iediately iplies (2)! 4 (!) 2 > 2 3π 1, for all N. Therefore, followig the lies i the proof of Lea 2.3 i 6], case (i), we iediately obtai ( ) j p > e 3π =. 3πe Siilarly, followig the lies i the proof of Lea 2.3 i 6], case (ii), we get ( ) p,1 = (2 1)! ( 1 ) > 3π = π Cobiig the cases (i) ad (ii) i the proof of Lea 2.3 i 6], sice 2 3πe > 1 6π, it follows that the costat c 2 i the stateet of Lea 2.3 i 6] ca be chose as c 2 = 1 6π. I coclusio, goig back with the values of the costats, we obtai c 1 = 1 6π 1 e ad 1 = 6πe 2, which fiish the proof. c 2 1 Also, we preset: Lea 2.3. For ay f C + 0, 1] ad N we have B (f)](x) B (f)(x), for all x 0, 1]. Proof. Let us choose arbitrary j {0, 1,..., }. By relatio (4.17) i 3], oe has B (f)(x) = f k,,j (x), x j/( + 1), (j + 1)/( + 1)], (1) where ( k) f k,,j (x) = ( j) ( ) k j x f(k/) 1 x for all k {0, 1,..., }. Relatio (1) iplies B (f)(x) f k,,j (x) for all x j/( + 1), (j + 1)/( + 1)] ad k {0, 1,..., }. I particular, for x = j/ j/( + 1), (j + 1)/( + 1)] ad k = j, we get B (f)(j/) f j,,j (j/) = f(j/), j {0, 1,..., }. Therefore, takig ito accout the relatioship of defiitio for B (f)(x) i Itroductio, we iediately get the stateet of the lea. We are ow i positio to prove the first ai result of this sectio.
5 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 43 Theore 2.4. For a fixed f C + 0, 1], let us cosider the iterative sequece of successive approxiatios a () (f)(x) = B ] (f)(x),, N, x 0, 1]. Here B ] 2 (f)(x) = B B (f)](x) ad so o. (i) For ay fixed N, there exists f : 0, 1] R +, such that f C + 0, 1], f Lip L 1 with L = 6πe 2 2 f, f (0) = f(0), f (1) = f(1), B B li + a() (f) = f, uiforly i 0, 1], (f )(x) = f (x) for all x 0, 1] (that is f is a fixed poit for the operator ) ad (f)(x) = a () 1 (f)(x) a() (f)(x) a () +1 (f)(x) f (x) f, for all x 0, 1], N; (ii) For all, N ad x 0, 1], we have the estiate ] (f)(x) f(x) 12 ω 1 (f; ), + 1 where ω 1 (f; δ) = sup{ f(x) f(y) ; x y δ}; (iii) For ay fixed N we have li a () (f)(x) = f(x), uiforly i 0, 1]; (iv) Let = depedig o such that li = 0. The we have li a () (f)(x) = f(x), uiforly i 0, 1]; (v) Suppose, i additio, that f Lip L 1 ad that it is strictly positive o 0, 1]. The, for all, N we have the estiate B ] (f) f ( ) L L + 4, f where f = if{f(x); x 0, 1]} > 0; (vi) Suppose that f Lip L 1 ad that it is strictly positive o 0, 1]. Let = depedig o such that li = 0. The uiforly o 0, 1] we have li a () (f)(x) = f(x). (vii) Suppose that f C + 0, 1] is such that for ay N, the fuctio B (f) is a fixed poit for the operator B. The, for ay sequece of atural ubers, ( ) N, the sequece of iterates a () (f) = ] (f) coverges uiforly o 0, 1] to f, as. Proof. (i) Fro the above Lea 2.3, easily follow the iequalities 0 (f)(x) = a () 1 (f)(x)... a() (f)(x) a () +1 (f)(x)... f, for all, N. The last iequality follows fro the obvious iequality 0 B (f)(x) f. Fixig N ad x 0, 1], the sequece of positive ubers (a () (f)(x)) N is bouded ad ootoically odecreasig, which iplies, for +, its covergece to a liit, deote it by f (x). Sice B (f)(x) f, we easily obtai a () (f)(x) f, for all, that is the sequece (a () (f)) N is uiforly bouded. Passig to liit with + we get f (x) f for all x 0, 1], N.
6 44 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN Also, sice it is easy to check that it is iediate that a () (f)(0) = f(0) ad a () therefore iplies that f (0) = f(0), f (1) = f(1). (f)(0) = f(0) ad B (f)(1) = f(1), (f)(1) = f(1) for all N, which Now, fro B (f) f ad applyig successively Theore 2.2, we easily obtai that a () (f) = B ] (f) Lip L 1, for all N. Therefore, the sequece (of fuctios of successive approxiatio) (a () (f)) N clearly is equicotiuous, which cobied with the fact that the sequece is uiforly bouded, by the Arzela- Ascoli theore iplies that it cotais a subsequece (a () k (f)) k N, uiforly coverget. Because the whole sequece is poitwise coverget to f (x), we get that li k a () k (f) = f uiforly i 0, 1] ad as a cosequece, it iediately follows that f C + 0, 1], i fact oreover, that f Lip L 1 with L = 6πe 2 2 f. Applyig ow the well-kow Dii s theore to the poitwise coverget ootoe sequece of cotiuous fuctios (a () (f)) N, it follows that i fact we have li a () (f) = f uiforly i 0, 1]. Also, the ootoicity of the sequece (a () ) N iplies a () all x 0, 1],, N. Fially, sice a () +1 (f) = B a () (f)] ad li a () 0, 1], takig also ito accout that by Theore 2.1, B fixed it follows that for all N we have (f)(x) f (x) for +1 (f) = f uiforly i is oexpasive, for ay (f ) f B (f ) a () +1 (f) + a() +1 (f) f f a () (f) + a () +1 (f) f. Passig here with, we get B (f ) f = 0, that is B (f )(x) f (x) = 0, for all x 0, 1]. (ii) For ay fixed N ad N variable, it is easy to see that the sequece ] (f)) N satisfies the Corollary 2.4 i 3], that is for all δ > 0 we get B ] (f)(x) f(x) ] δ B ] (ϕ x )(x) ω 1 (f; δ), x 0, 1], ( where ϕ x (t) = t x, for all t 0, 1]. I what follows we prove by atheatical iductio that B ] (ϕ x )(x) 6 +1, for all, N, x 0, 1], which replaced i the above estiate ad by choosig the δ = 6 +1, will iediately iply ) B ] (f)(x) f(x) 12 ω 1 (f;. + 1 Ideed, deotig ( k) k,,j (x) = ( j) ( ) k j x, 1 x
7 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 45 by 3], relatioship (4.17), we ca write ( ) k B (f)(x) = k,,j (x)f, for all x j/( + 1), (j + 1)/( + 1)]. This iediately iplies ] 2 (f)(x) = = k,,j (x)b (f)(k/) ] k,,j (x) i,,k (k/)f(i/). i=0 Replacig here f(t) = t x = ϕ x (t) with x fixed, ad takig ito accout the iequality i x i k + k x, for all x j/( + 1), (j + 1)/( + 1)] we get ] B ] 2 (ϕ x )(x) = k,,j (x) i,,k (k/) i x i=0 k,,j (x) i,,k (k/) k i ] i=0 ] + k,,j (x) i,,k (k/) k x i=0 = k,,j (x) i,,k (k/) k i ] i=0 + k,,j (x) k x ] i,,k (k/) i= = For the last estiate we used the iequalities which follow fro the relatioship (4.6) i the proof of Theore 4.1 i 3] k,,j (x) k x 6, i,,k (k/) k + 1 i ad the iequalities obtaied fro Lea 3.2 i 3] k,,j (x) 1, i,,k (k/) 1. Siilarly, takig ito accout that for all x j/( + 1), (j + 1)/( + 1)] we ca write B ] 3 (f)(x)
8 46 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN = ]] k,,j (x) i,,k (k/) l,,i (i/)f(l/), i=0 replacig here f(t) = t x = ϕ x (t), takig ito accout the iequality l x l i + i k + k x, ad reasoig exactly as i the case of B ] 2, we easily obtai B ] 3 3 (ϕ x )(x) 6, x j/( + 1), (j + 1)/( + 1)], + 1 valid for all j = 0, 1,...,. Therefore, the above iequality is i fact valid for all x 0, 1]. Reasoig ow by atheatical iductio, we get the desired estiate i the stateet for arbitrary N. (iii) It is iediate by passig to liit with i the iequality fro the above poit (ii). (iv) It is iediate by replacig with i the estiate i (ii), by passig to liit with ad takig ito accout that li +1 = 0. (v) We obviously ca write B ] (f) f B ] j (f) B ] j 1 (f), where by covetio B ] 0 (f)(x) = f(x). But by applyig successively Theore 2.1, we easily get that j=1 B ] j (f) B ] j 1 (f) B ] j 1 (f) B ] j 2 (f) (... B ](f) (f) ω 1 f; 1 ) ] ω1 (f; 1/) + 4, f where for the last estiate above we used Theore 4.6 i 5], valid for strictly positive fuctios oly. Now, takig ito accout that f Lip L 1, fro the above estiate we get l=0 ] j (f) B ] j 1 (f) 1 for all j = 1,...,, which fially iplies B ] (f) f L ( L f + 4 ( )] L L + 4. f )], (vi) It is iediate by takig = ad passig to liit i the estiate fro the above poit (v). (vii) By hypothesis, we have B B (f)] = B (f), for all N, ad therefore it easily follows that B ] (f) = B (f), for all N. Cosequetly, by Theore 4.1 i 3], we obtai B ] (f)(x) f(x) = B (f)(x) f(x) 12 ω 1 (f; 1/ + 1),
9 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 47 ad passig to liit with, we iediately get the desired coclusio. Rearks. 1) I the class of Lipschitz, strictly positive fuctios, Theore 2.4, (vi), is ore geeral tha Theore 2.4, (iv). Ideed, while li = 0 iplies li = 0, the coverse is ot true. Note that the case of Theore 2.4, (vi), is siilar with what happes i the case of the iterates of Berstei polyoials. 2) As a cosequece of the well-kow Trotter s approxiatio result i the theory of the seigroups of liear operators (see e.g. 9]), it is kow that i the case of Berstei polyoials B (f)(x), if f is twice differetiable ad li = t > 0, the li B (f)(x) = e ta(x), where A(x) = x(1 x)f (x) 2, for all x 0, 1]. It reais as a iterestig ope questio what happes with the iterates ] (f), whe li = t > 0. Let us first observe that by Theore 2.4, (vii), if f satisfies the hypothesis there, the B ] (f) uiforly coverges to f o 0, 1]. It is worth etioig that by the ext Theores 2.5 ad 2.6, we put i evidece large classes of fuctios f satisfyig the hypothesis i Theore 2.4, (vii). Therefore, the above etioed ope proble for the Berstei ax-product operator, gets a sese oly if f does ot satisfy the hypothesis i Theore 2.4, (vii). Also, otice here that the Berstei ax-product operator B ] is ot liear. 3) If f is a fixed poit of B (f)(x) = B, i.e. f(x) = B (f)(x) for all x 0, 1], we easily (f)(x), for all N, x 0, 1], therefore i this case it is get a () trivial i Theore 2.4, (i), that f (x) = B (f)(x), for all x 0, 1]. 4) Accordig to Theore 2.4, (i), for each fixed N it is iportat to deterie the set of the fixed poits for B. I this sese, we preset the followig results. Theore 2.5. (i) If f : 0, 1] 0, ) is odecreasig ad such that the fuctio g : (0, 1] 0, ), g(x) = f(x) x is oicreasig, the for ay N, B (f) is a fixed poit for the operator B, that is B B (f)](x) = B (f)(x), for all x 0, 1]; (ii) If f : 0, 1] 0, ) is oicreasig ad such that the fuctio h : 0, 1) 0, ), h(x) = f(x) 1 x is odecreasig, the for ay N, B (f) is a fixed poit for the operator, that is (f)](x) = (f)(x), for all x 0, 1]. Proof. (i) Fro the relatios (4.46) ad (4.47) i the proof of Corollary 4.7 i 3], for all x j/( + 1), (j + 1)/( + 1)] ad j {0, 1,..., 1} we ca write ad where B (f)(x) = ax{f j,,j (x), f j+1,,j (x)} (f)(x) = f(1), for x /( + 1), 1], ( k) f k,,j (x) = ( j) ( ) k j x f(k/). 1 x Takig above x = j/, by siple calculatio we obtai (f)(j/) = ax{f(j/), f(j + 1)/] j/(j + 1)},
10 48 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN which by the property of the auxiliary fuctio g i hypothesis, iplies f(j/) f(j + 1)/], which replaced i the above equality gives B (f)(j/) = f(j/). j j+1 But it is clear that if for f C + 0, 1] we have B (f)(j/) = f(j/) for all j {0, 1,..., }, the g = B (f) is a fixed poit for B, which iplies the desired coclusio. (ii) Fro the relatios (4.49) ad (4.50) i the proof of Corollary 4.7 i 3], for all x j/( + 1), (j + 1)/( + 1)] ad j {1,..., } we ca write B (f)(x) = ax{f j 1,,j (x), f j,,j (x)}, ad B (f)(x) = f(0), for x 0, 1/( + 1)]. Takig above x = j/, by siple calculatio we obtai B (f)(j/) = ax{f(j 1)/] ( j)/( j + 1), f(j/)}, which by the property of the auxiliary fuctio g i hypothesis, iplies f(j/) j j+1f(j 1)/], which replaced i the above equality gives B (f)(j/) = f(j/). Therefore, we agai get the desired coclusio. Rearks. 1) Accordig to Reark 4.8 i 3], if f : 0, 1] 0, ) is a covex, odecreasig fuctio satisfyig f(x) x f(1) for all x 0, 1], or if f : 0, 1] 0, ) is a covex, oicreasig fuctio satisfyig f(x) 1 x f(0), the agai f satisfies the hypothesis i Theore 2.5, (i) ad (ii), respectively, ad cosequetly we get (f)](x) = B (f)(x), for all x 0, 1]. 2) Deote by S0, 1] the class of all fuctios f which satisfy the hypothesis i the stateet of Theore 2.5 (i), or of Theore 2.5 (ii), or i the above Reark 1. Also, for ay fixed arbitrary N, let us deote T 0, 1] = B (S0, 1]) = {F C + 0, 1]; f S0, 1] such that F (x) = (f)(x), x 0, 1]}. The if we deote by F 0, 1] = {F : 0, 1] 0, + ); B (F )(x) = F (x), for all x 0, 1]} : C + 0, 1] C + 0, 1], the stateet 0, 1] F 0, 1]. 3) By Lea 4.6 i 3], ay odecreasig cocave fuctio satisfies the hypothesis of Theore 2.5, (i), ad ay oicreasig cocave fuctio satisfies the hypothesis the set of all fixed poits of the operator of Theore 2.5 together with the above Reark 1 eas that we have T of Theore 2.5, (ii). Therefore, the class of all positive, ootoe ad cocave fuctios o 0, 1] deoted by MK + 0, 1], has the property MK + 0, 1] S0, 1], that is the fuctio H = B (f) satisfies B (H)(x) = H(x), for all x 0, 1]. 4) It is easy to cosider cocrete exaples of fuctios i S0, 1] (others tha the costat fuctios which obviously are fixed poits for ), like x, e x, 1 + x 2, si(x), cos(x), l(1 + x), e x, 1 + x 3.
11 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 49 Ideed, it is easy to check that x, e x ad 1 + x 2 satisfy the first type of hypothesis i the above Reark 1, si(x), cos(x) ad l(1 + x) belog to the class MK + 0, 1] defied i the above Reark 3, while e x satisfy the secod type of hypothesis i the above Reark 1. Therefore, for ay f i this reark we have B B (f)](x) = B (f)(x), for all x 0, 1] ad N. The results expressed by the above Reark 3 ca be geeralized to the whole class of cocave fuctios, as follows. Theore 2.6. If f : 0, 1] 0, ) is a cotiuous cocave fuctio the we have ] (f) = (f) for all N. Proof. By the proof of Corollary 4.6. i 3] we get B (f)(x) = ax{f j 1,,j (x), f j,,j (x), f j+1,,j (x)} for all x j/( + 1), (j + 1)/( + 1)] ad j {1,..., 1}, ad Here recall that (f)(x) = ax{f 0,,0 (x), f 0,,1 (x)} for all x 0, 1/( + 1)] (f)(x) = ax{f,, 1 (x), f,, (x)}, for all x /( + 1), 1]. ( k) f k,,j (x) = ( j) ( ) k j x f(k/). 1 x Sice j/ j/( + 1), (j + 1)/( + 1)], replacig x = j/ i the above forulas for B (f)(x), we easily obtai (see the reasoigs i the proof of Theore 2.5, (i) ad (ii)) that B (f)(j/) = f(j/) for all j {0, 1,..., }, which for the forula of defiitio of B (f)(x) easily iplies the desired coclusio. Rearks. 1) Theores 2.5 ad 2.6 put i evidece large classes of fuctios f C + 0, 1], with the property that B (f) is a fixed poit for the operator B, for all N. The followig exaple of f is that of a fuctio for which there exists N (i fact a ifiity of such of ) such that B (f) is ot ayore fixed poit for the operator B. Ideed, let f : 0, 1] 0, ) be defied by f(x) = 1/2 x if x 0, 1/2] ad f(x) = x 1/2 if x (1/2, 1]. For = 5, by the forula of defiitio of B (f)(x) i Itroductio, we easily get ad 5 (f)(0) = 5 (f)(1) = 1/2, 5 (f)(1/5) = 5 (f)(4/5) = 2/5, 5 (f)(2/5) = 5 (f)(3/5) = 9/40, 5 ( 5 (f))(2/5) = 3/10. Therefore, it follows 5 ( 5 (f))(2/5) 5 (f)(2/5), which clearly iplies that 5 (f) is ot a fixed poit for the operator 5.
12 50 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN I fact, by usig for exaple MATLAB, oe ca easily show that for ay other values of (sufficietly large), agai we get the sae coclusio. 2) Theore 2.6 is also useful to show that the ethod i the case of Berstei polyoials i 11] caot be use here, because for ay a, b R +, the operator B caot be a cotractio o the subspace U a,b = {f C + 0, 1]; f(0) = a, f(1) = b}. I this sese, we ca prove that for ay atural uber, there exist two cotiuous fuctios f, g : 0, 1] 0, ) satisfyig f(0) = g(0) = a, f(1) = g(1) = b ad such that B (f) B (g) = f g. Ideed, let us defie as y = f(x) the equatio of the straight lie passig through the poits (0, a) ad (1, b) ad let g be the fuctio whose graph is the polygoal lie passig through the poits (0, a), (1/2, c) ad (1, b), where the value c ca be ay real uber which satisfies c > f(1/2). (Note that the graphs of both fuctios f ad g for a triagle.) By siple geoetrical reasoigs we get that f g = g(1/2) f(1/2). Firstly, we suppose that is eve. Sice f ad g are cocave fuctios, by the proof of the above Theore 2.6, we get B (f)(j/) = f(j/) ad siilarly, B (g)(j/) = g(j/) for all j {0, 1,..., }. Therefore, takig j() = /2, we obtai that B (f)(1/2) = f(1/2) ad B (g)(1/2) = g(1/2). I coclusio, we have g(1/2) f(1/2) = f g B (f) B (g) B (f)(1/2) B (g)(1/2) = g(1/2) f(1/2), which iplies B (f) B (g) = f g, for ay eve atural uber. The reasoig is siilar i the case whe is ad odd atural uber, because it suffices to replace the pair (1/2, c) i the defiitio of g with ( 0 /( ), c) where = Ishikawa Iteratios for The results i this sectio are based o the followig two well-kow results. Theore 3.1. (Ishikawa 8]) Let C be a copact covex subset of a Baach space (X, ) ad T : C C be oexpasive. For (λ ) N a sequece i 0, b] with b < 1 ad such that =0 λ = +, let us defie the iterates i X by x +1 := (1 λ )x + λ T (x ). The for ay startig poit x 0 C, the sequece (x ) N coverges to a fixed poit of T. Theore 3.2. (Ishikawa 8]) Let C be a closed bouded covex subset of a Baach space (X, ) ad T : C C be oexpasive. Let (λ ) be as i Theore 3.1. The for ay startig poit x 0 C, the followig sequece, ( x T (x ) ) N, coverges to 0 (i.e. (x ) is a so-called approxiate fixed-poit sequece). Now, i order to ca apply to our case the above Theores 3.1 ad 3.2, firstly we eed to idetify bouded closed covex ad copact covex subsets i C + 0, 1]. For
13 ITERATIONS OF THE BERNSTEIN MAX-PRODUCT OPERATOR 51 exaple, it is easy to check that the subset C + K 0, 1] = {f C +0, 1]; f K}, is bouded, closed ad covex. Also, it is easy to check that the subset C L,K = C + K 0, 1] Lip L 1 is bouded, closed, covex ad equicotiuous, which by the Arzela- Ascoli theore iplies that C L,K is a covex copact subset i C + 0, 1] edowed with the uifor or. Aother iportat hypothesis i the Theores 3.1 ad 3.2 is the ivariace property of T. I our case, we eed this ivariace property for the Berstei ax-product operator. For this purpose, we will ake use of the Theore 2.2 i Sectio 2. We have: Theore 3.3. (i) If f C + K 0, 1] the for all N we have B (f) C + K 0, 1]; (ii) Let K > 0 ad L 6πe 2 K be fixed costats ad deote C L,K = C + K 0, 1] Lip L 1. The, for all N satisfyig the iequality 2 L 6πe 2 K, the ivariace property B (C L,K ) C L,K holds. Proof. (i) Sice 0 f(k/) f for all N ad k = 0, 1,...,, it is iediate by the forula of defiitio of B (f)(x), because we easily get B (f)(x) f, for all x 0, 1], which iplies B f K, for all N. (ii) Let f C L,K. By (i) it follows that B (f) K for all N ad by (i) it follows that B (f) Lip 6πe 2 2 f 1 Lip 6πe2 2 K 1, for all N. The, by 2 we get B (f) Lip 6πe2 2 K 1 Lip L 1, which leads to the coclusio L 6πe 2 K 6πe 2 K. As iediate cosequeces of the above cosideratios, we get the followig two results. Corollary 3.4. Let K > 0 ad L 6πe 2 K be fixed costats ad C L,K = C + K 0, 1] Lip L 1. Also, let (λ ) N be sequece i 0, b] with b < 1 ad such that =0 λ = +. For ay N ad f,1 C L,K fixed, let us defie the iterated that (f) C L,K for satisfyig 2 L sequece of fuctios f,+1 (x) = (1 λ )f, (x) + λ (f, )(x), N, x 0, 1]. The, for ay fixed N satisfyig the iequality 2 L 6πe 2 K, the sequece of fuctios (f, (x)) N coverges as i the uifor or, to a fixed poit of the operator B. Proof. Firstly, it is clear that C + 0, 1] edowed with the uifor or is a Baach space. By Theore 2.1, by the coets betwee the stateets of the Theores 3.2 ad 3.3 ad by Theore 3.3, (ii), the operator B : C L,K C L,K is oexpasive o the copact covex set C L,K. The the corollary is a iediate cosequece of Theore 3.1. Corollary 3.5. Let K > 0 ad C + K 0, 1] = {f C +0, 1]; f K}. Also, let (λ ) ad the iterated sequece (f,+1 (x)) N be defied as i the stateet of Corollary 3.4. The, for ay N ad f,1 C + K 0, 1] fixed, we have li f, (f, ) = 0,
14 52 MIRCEA BALAJ, LUCIAN COROIANU, SORIN G. GAL AND SORIN MUREŞAN where deotes the uifor or. Proof. By Theore 2.1, by the coets betwee the stateets of the Theores 3.2 ad 3.3 ad by Theore 3.3, (i), the operator B : C + K 0, 1] C+ K 0, 1] is oexpasive o the bouded, closed ad covex subset C + K 0, 1]. The the corollary is a iediate cosequece of Theore 3.2. Reark. The ethods i this paper ca be exteded to other ax-product operators of Berstei-type. Ackowledgeet. The work of all authors was supported by a grat of the Roaia Natioal Authority for Scietific Research, CNCS UEFISCDI, project uber PN-II-ID-PCE Also, the work of the secod author was supported by the Sectoral Operatioal Prograe for Hua Resources Developet , co-fiaced by the Europea Social Fud, uder the project uber POSDRU/-/107/1.5/S/76841 with the title Moder Doctoral Studies:Iteratioalizatio ad Iterdiscipliarity. Refereces 1] O. Agratii, O soe Berstei type operators: iterates ad geeralizatios, East J. Approx., 9(2003), o. 4, ] O. Agratii, I.A. Rus, Iterates of a class of discrete liear operators via cotractio priciple, Coet. Math. Uiv. Caroli., 44(2003), o. 3, ] B. Bede, L. Coroiau, S.G. Gal, Approxiatio ad shape preservig properties of the Berstei operator of ax-product kid, Iter. J. Math. ad Math. Sci., volue 2009, Article ID , 26 pages, doi: /2009/ ] B. Bede, S.G. Gal, Approxiatio by oliear Berstei ad Favard-Szász-Mirakja operators of ax-product kid, Joural of Cocrete ad Applicable Matheatics, 8(2010), o. 2, ] L. Coroiau, S.G. Gal, Classes of fuctios with iproved estiates i approxiatio by the ax-product Berstei operator, Aalysis ad Applicatios, 9(2011), o. 3, ] L. Coroiau, S.G. Gal, Global soothess preservatio by soe oliear ax-product operators, Mateaticki Vesik, (i press), 7] S.G. Gal, Shape-Preservig Approxiatio by Real ad Coplex Polyoials, Birkhäuser, Bosto-Basel-Berli, ] S. Ishikawa, Fixed poits ad iteratios of a oexpasive appig i a Baach space, Proc. Aer. Math. Soc., 59(1976), ] S. Karli, Z. Ziegler, Iteratio of positive approxiatio operators, J. Approx. Theory, 3(1970), ] R.P. Kelisky, T.J. Rivli, Iterates of Berstei polyoials, Pacific J. Math., 21(1967), ] I.A. Rus, Iterates of Berstei operators, via cotractio priciple, J. Math. Aal. Appl., 292(2004), ] I.A. Rus, Iterates of Stacu operators (via fixed poit theory priciples) revisited, Fixed Poit Theory, 11(2010), No. 2, Received: Jauary 30, 2012; Accepted: March 15, 2012.
A New Type of q-szász-mirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationA GENERALIZED BERNSTEIN APPROXIMATION THEOREM
Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0029-x Tatra Mt. Math. Publ. 49 2011, 99 109 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationLocal Approximation Properties for certain King type Operators
Filomat 27:1 (2013, 173 181 DOI 102298/FIL1301173O Published by Faculty of Scieces ad athematics, Uiversity of Niš, Serbia Available at: http://wwwpmfiacrs/filomat Local Approimatio Properties for certai
More informationBertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationApproximation by max-product type nonlinear operators
Stud. Uiv. Babeş-Bolyai Math. 5620, No. 2, 34 352 Approximatio by max-product type oliear operators Sori G. Gal Abstract. The purpose of this survey is to preset some approximatio ad shape preservig properties
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationA new sequence convergent to Euler Mascheroni constant
You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS
It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,
More informationAutomated Proofs for Some Stirling Number Identities
Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationSZEGO S THEOREM STARTING FROM JENSEN S THEOREM
UPB Sci Bull, Series A, Vol 7, No 3, 8 ISSN 3-77 SZEGO S THEOREM STARTING FROM JENSEN S THEOREM Cǎli Alexe MUREŞAN Mai îtâi vo itroduce Teorea lui Jese şi uele coseciţe ale sale petru deteriarea uǎrului
More informationTHE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION
MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia
More informationSome remarks on the paper Some elementary inequalities of G. Bennett
Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries
More informationChapter 2. Asymptotic Notation
Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It
More informationBERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS
STUDIA UNIV. BABEŞ BOLYAI MATHEMATICA Volue LIV Nuber 4 Deceber 2009 BERNSTEIN-TYPE OPERATORS ON TETRAHEDRONS PETRU BLAGA TEODORA CĂTINAŞ AND GHEORGHE COMAN Abstract. The ai of the paper is to costruct
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More informationDISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES
MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS Vol, No, 4, pp5- http://doiorg/88/ep4 DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES Yogchao Hou* ad Weicai Peg Departet of Matheatical Scieces, Chaohu Uiversity,
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 4, December 2002 ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS OGÜN DOĞRU Dedicated to Professor D.D. Stacu o his 75
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationA Study on the Rate of Convergence of Chlodovsky-Durrmeyer Operator and Their Bézier Variant
IOSR Joural of Matheatics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volue 3, Issue Ver. III (Mar. - Apr. 7), PP -8 www.iosrjourals.org A Study o the Rate of Covergece of Chlodovsky-Durreyer Operator ad
More informationANSWERS TO MIDTERM EXAM # 2
MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationThe Hypergeometric Coupon Collection Problem and its Dual
Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther
More informationStrong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types
It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics
More informationII. EXPANSION MAPPINGS WITH FIXED POINTS
Geeralizatio Of Selfmaps Ad Cotractio Mappig Priciple I D-Metric Space. U.P. DOLHARE Asso. Prof. ad Head,Departmet of Mathematics,D.S.M. College Jitur -431509,Dist. Parbhai (M.S.) Idia ABSTRACT Large umber
More informationA PROBABILITY PROBLEM
A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,
More informationDouble Derangement Permutations
Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri
More informationJORGE LUIS AROCHA AND BERNARDO LLANO. Average atchig polyoial Cosider a siple graph G =(V E): Let M E a atchig of the graph G: If M is a atchig, the a
MEAN VALUE FOR THE MATCHING AND DOMINATING POLYNOMIAL JORGE LUIS AROCHA AND BERNARDO LLANO Abstract. The ea value of the atchig polyoial is coputed i the faily of all labeled graphs with vertices. We dee
More information1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND
NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss
More informationSOME GENERALIZATIONS OF OLIVIER S THEOREM
SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers.
More informationTHE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA
Global Joural of Advaced Research o Classical ad Moder Geometries ISSN: 2284-5569, Vol.6, 2017, Issue 2, pp.119-125 THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA DAN
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationOn the Variations of Some Well Known Fixed Point Theorem in Metric Spaces
Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow
More informationConcavity of weighted arithmetic means with applications
Arch. Math. 69 (1997) 120±126 0003-889X/97/020120-07 $ 2.90/0 Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Cocavity of weighted arithmetic meas with applicatios By ARKADY BERENSTEIN ad ALEK VAINSHTEIN*)
More informationMath 140A Elementary Analysis Homework Questions 3-1
Math 0A Elemetary Aalysis Homework Questios -.9 Limits Theorems for Sequeces Suppose that lim x =, lim y = 7 ad that all y are o-zero. Detarime the followig limits: (a) lim(x + y ) (b) lim y x y Let s
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationRefinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane
Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex
More information2 Banach spaces and Hilbert spaces
2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationSOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER
Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationApproximation by Superpositions of a Sigmoidal Function
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 22 (2003, No. 2, 463 470 Approximatio by Superpositios of a Sigmoidal Fuctio G. Lewicki ad G. Mario Abstract. We geeralize
More informationarxiv: v1 [math.nt] 26 Feb 2014
FROBENIUS NUMBERS OF PYTHAGOREAN TRIPLES BYUNG KEON GIL, JI-WOO HAN, TAE HYUN KIM, RYUN HAN KOO, BON WOO LEE, JAEHOON LEE, KYEONG SIK NAM, HYEON WOO PARK, AND POO-SUNG PARK arxiv:1402.6440v1 [ath.nt] 26
More informationMetric Space Properties
Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationCommon Fixed Points for Multifunctions Satisfying a Polynomial Inequality
BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 60-65 Seria Mateatică - Iforatică - Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea Petrol-Gaze
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More information19.1 The dictionary problem
CS125 Lecture 19 Fall 2016 19.1 The dictioary proble Cosider the followig data structural proble, usually called the dictioary proble. We have a set of ites. Each ite is a (key, value pair. Keys are i
More informationOn Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings
Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad
More informationInternational Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN
Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationREVIEW OF CALCULUS Herman J. Bierens Pennsylvania State University (January 28, 2004) x 2., or x 1. x j. ' ' n i'1 x i well.,y 2
REVIEW OF CALCULUS Hera J. Bieres Pesylvaia State Uiversity (Jauary 28, 2004) 1. Suatio Let x 1,x 2,...,x e a sequece of uers. The su of these uers is usually deoted y x 1 % x 2 %...% x ' j x j, or x 1
More informationDifferentiable Convex Functions
Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for
More informationAN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION
Joural of Statistics: Advaces i Theory ad Applicatios Volue 3, Nuber, 00, Pages 6-78 AN EFFICIENT ESTIMATION METHOD FOR THE PARETO DISTRIBUTION Departet of Matheatics Brock Uiversity St. Catharies, Otario
More informationSOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR
Joural of the Alied Matheatics Statistics ad Iforatics (JAMSI) 5 (9) No SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR SP GOYAL AND RAKESH KUMAR Abstract Here we
More informationWeighted Approximation by Videnskii and Lupas Operators
Weighted Approximatio by Videsii ad Lupas Operators Aif Barbaros Dime İstabul Uiversity Departmet of Egieerig Sciece April 5, 013 Aif Barbaros Dime İstabul Uiversity Departmet Weightedof Approximatio Egieerig
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationApproximation of fuzzy numbers by nonlinear Bernstein operators of max-product kind
EUSFLAT-LFA 20 July 20 Aix-les-Bais, Frace Approximatio of fuzzy umbers by oliear Berstei operators of max-product kid Lucia Coroiau Sori G. Gal 2 Barabás Bede 3,2 Departmet of Mathematics ad Computer
More informationGeneralized Fixed Point Theorem. in Three Metric Spaces
It. Joural of Math. Aalysis, Vol. 4, 00, o. 40, 995-004 Geeralized Fixed Poit Thee i Three Metric Spaces Kristaq Kikia ad Luljeta Kikia Departet of Matheatics ad Coputer Scieces Faculty of Natural Scieces,
More informationq-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time.
-Fiboacci polyoials ad -Catala ubers Joha Cigler The Fiboacci polyoials satisfy the recurrece F ( x s) = s x = () F ( x s) = xf ( x s) + sf ( x s) () with iitial values F ( x s ) = ad F( x s ) = These
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationLebesgue Constant Minimizing Bivariate Barycentric Rational Interpolation
Appl. Math. If. Sci. 8, No. 1, 187-192 (2014) 187 Applied Matheatics & Iforatio Scieces A Iteratioal Joural http://dx.doi.org/10.12785/ais/080123 Lebesgue Costat Miiizig Bivariate Barycetric Ratioal Iterpolatio
More informationSequences and Series
Sequeces ad Series Sequeces of real umbers. Real umber system We are familiar with atural umbers ad to some extet the ratioal umbers. While fidig roots of algebraic equatios we see that ratioal umbers
More informationAN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS. Robert DEVILLE and Catherine FINET
2001 vol. XIV, um. 1, 95-104 ISSN 1139-1138 AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS Robert DEVILLE ad Catherie FINET Abstract This article is devoted to a extesio of Simos iequality. As a cosequece,
More informationUniversity of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!
Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad
More informationLecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009
18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationAppendix to Quicksort Asymptotics
Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationOn Some Properties of Tensor Product of Operators
Global Joural of Pure ad Applied Matheatics. ISSN 0973-1768 Volue 12, Nuber 6 (2016), pp. 5139-5147 Research Idia Publicatios http://www.ripublicatio.co/gjpa.ht O Soe Properties of Tesor Product of Operators
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationA NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p
A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p SEICK KIM Abstract. Assume that Ω is a bouded domai i R with 2. We study positive solutios to the problem, u = u p i Ω, u(x) as x Ω, where p > 1. Such solutios
More informationAVERAGE MARKS SCALING
TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I
More information